pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, spdep, GWmodel, tmap, skimr, caret)In-class Exercise 5: Modeling the Spatial Variation of the Explanatory Factors of Water Point Status using Geographically Weighted Logistic Regression
Setting the scene
To build an exploratory model to discover factors affecting water point in Osun State, Nigeria
Study area: Osun State, Nigeria
Data sets used:
Osun.rds, which contains LGA boundaries of Osun state. It is in sf polygon data frame, and
Osun_wp_sf.rds, which contains water points within Osun state. It is in sf point data frame
Model Variables
Dependent variable: Water point status (i.e. functional/non-functional)
Independent variables:
distance_to_primary_road (continuous),
distance_to_secondary_road (continuous),
distance_to_tertiary_road (continuous),
distance_to_city (continuous),
distance_to_town (continuous),
water_point_population (continuous),
local_population_11km (continuous),
usage_capacity (categorical),
is_urban (categorical),
water_source_clean (categorical)
IntroductionLogistic Regression Model
We will take a look at the theory of Logistic regression model before performing our analysis.
Simple Logistic Regression Model
Let’s take a look at the formula for a simple logistic regression model:
Multiple Logistic Regression model
Assumptions of logistic regression
Logistic regression does not assume a linear relationship between dependent and independent variables.
For binary logistic regression, the dependent variable must be a dichotomy (2 categories).
The independent variables need not be interval, nor normally distributed, nor linearly related, nor of equal variance within each group.
The categories (groups) must be mutually exclusive and exhaustive; a case can only be in one group and every case must be a member of one of the groups.
Data requirements of logistic regression
Large samples are required
It does not use least squares method, but instead, it uses an algorithm called maximum likelihood (expectation of the probability)
Getting started
The code chunks below installs and launches the R packages into R environment. The below packages mentioned are specifically used for this exercise to support our analysis.
blorr - for error matrix computations
skimr, funModeling - for exploratory data analysis
caret - commonly for machine learning purposes, and it is used in this exercise to compute the error matrix for comparison with the output from blorr.
Data import
We will now import the required dataset into the R environment. Two datasets will be used.
Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")We will also use funModeling to take a look at the data records under the status column.
Osun_wp_sf %>% freq(input="status")Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
Please report the issue at <]8;;https://github.com/pablo14/funModeling/issueshttps://github.com/pablo14/funModeling/issues]8;;>.
status frequency percentage cumulative_perc
1 TRUE 2642 55.5 55.5
2 FALSE 2118 44.5 100.0tmap_mode("view")tmap mode set to interactive viewingtm_shape(Osun) +
# tmap_options(check.and.fix=TRUE) +
tm_polygons(alpha = 0.4) +
tm_shape(Osun_wp_sf) +
tm_dots(col="status", alpha = 0.6) +
tm_view(set.zoom.limits = c(9,12))It is interesting to note here that the FALSE variables are closely clustered in some of the regions.
Exploratory Data Analysis (EDA)
Note: Regression models are very sensitive to missing values. Hence, we should remove all missing values from the data set before continuing with our analysis.
We will now use skimr to do EDA. It provides a tabular format of the summary statistics of the variables that we will be looking into.
Summary statistics with skimr
Osun_wp_sf %>% skim()Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.| Name | Piped data |
| Number of rows | 4760 |
| Number of columns | 75 |
| _______________________ | |
| Column type frequency: | |
| character | 47 |
| logical | 5 |
| numeric | 23 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| source | 0 | 1.00 | 5 | 44 | 0 | 2 | 0 |
| report_date | 0 | 1.00 | 22 | 22 | 0 | 42 | 0 |
| status_id | 0 | 1.00 | 2 | 7 | 0 | 3 | 0 |
| water_source_clean | 0 | 1.00 | 8 | 22 | 0 | 3 | 0 |
| water_source_category | 0 | 1.00 | 4 | 6 | 0 | 2 | 0 |
| water_tech_clean | 24 | 0.99 | 9 | 23 | 0 | 3 | 0 |
| water_tech_category | 24 | 0.99 | 9 | 15 | 0 | 2 | 0 |
| facility_type | 0 | 1.00 | 8 | 8 | 0 | 1 | 0 |
| clean_country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| clean_adm1 | 0 | 1.00 | 3 | 5 | 0 | 5 | 0 |
| clean_adm2 | 0 | 1.00 | 3 | 14 | 0 | 35 | 0 |
| clean_adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| clean_adm4 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| installer | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management_clean | 1573 | 0.67 | 5 | 37 | 0 | 7 | 0 |
| status_clean | 0 | 1.00 | 9 | 32 | 0 | 7 | 0 |
| pay | 0 | 1.00 | 2 | 39 | 0 | 7 | 0 |
| fecal_coliform_presence | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| subjective_quality | 0 | 1.00 | 18 | 20 | 0 | 4 | 0 |
| activity_id | 4757 | 0.00 | 36 | 36 | 0 | 3 | 0 |
| scheme_id | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| wpdx_id | 0 | 1.00 | 12 | 12 | 0 | 4760 | 0 |
| notes | 0 | 1.00 | 2 | 96 | 0 | 3502 | 0 |
| orig_lnk | 4757 | 0.00 | 84 | 84 | 0 | 1 | 0 |
| photo_lnk | 41 | 0.99 | 84 | 84 | 0 | 4719 | 0 |
| country_id | 0 | 1.00 | 2 | 2 | 0 | 1 | 0 |
| data_lnk | 0 | 1.00 | 79 | 96 | 0 | 2 | 0 |
| water_point_history | 0 | 1.00 | 142 | 834 | 0 | 4750 | 0 |
| clean_country_id | 0 | 1.00 | 3 | 3 | 0 | 1 | 0 |
| country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| water_source | 0 | 1.00 | 8 | 30 | 0 | 4 | 0 |
| water_tech | 0 | 1.00 | 5 | 37 | 0 | 20 | 0 |
| adm2 | 0 | 1.00 | 3 | 14 | 0 | 33 | 0 |
| adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management | 1573 | 0.67 | 5 | 47 | 0 | 7 | 0 |
| adm1 | 0 | 1.00 | 4 | 5 | 0 | 4 | 0 |
| New Georeferenced Column | 0 | 1.00 | 16 | 35 | 0 | 4760 | 0 |
| lat_lon_deg | 0 | 1.00 | 13 | 32 | 0 | 4760 | 0 |
| public_data_source | 0 | 1.00 | 84 | 102 | 0 | 2 | 0 |
| converted | 0 | 1.00 | 53 | 53 | 0 | 1 | 0 |
| created_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| updated_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| Geometry | 0 | 1.00 | 33 | 37 | 0 | 4760 | 0 |
| ADM2_EN | 0 | 1.00 | 3 | 14 | 0 | 30 | 0 |
| ADM2_PCODE | 0 | 1.00 | 8 | 8 | 0 | 30 | 0 |
| ADM1_EN | 0 | 1.00 | 4 | 4 | 0 | 1 | 0 |
| ADM1_PCODE | 0 | 1.00 | 5 | 5 | 0 | 1 | 0 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| rehab_year | 4760 | 0 | NaN | : |
| rehabilitator | 4760 | 0 | NaN | : |
| is_urban | 0 | 1 | 0.39 | FAL: 2884, TRU: 1876 |
| latest_record | 0 | 1 | 1.00 | TRU: 4760 |
| status | 0 | 1 | 0.56 | TRU: 2642, FAL: 2118 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| row_id | 0 | 1.00 | 68550.48 | 10216.94 | 49601.00 | 66874.75 | 68244.50 | 69562.25 | 471319.00 | ▇▁▁▁▁ |
| lat_deg | 0 | 1.00 | 7.68 | 0.22 | 7.06 | 7.51 | 7.71 | 7.88 | 8.06 | ▁▂▇▇▇ |
| lon_deg | 0 | 1.00 | 4.54 | 0.21 | 4.08 | 4.36 | 4.56 | 4.71 | 5.06 | ▃▆▇▇▂ |
| install_year | 1144 | 0.76 | 2008.63 | 6.04 | 1917.00 | 2006.00 | 2010.00 | 2013.00 | 2015.00 | ▁▁▁▁▇ |
| fecal_coliform_value | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| distance_to_primary_road | 0 | 1.00 | 5021.53 | 5648.34 | 0.01 | 719.36 | 2972.78 | 7314.73 | 26909.86 | ▇▂▁▁▁ |
| distance_to_secondary_road | 0 | 1.00 | 3750.47 | 3938.63 | 0.15 | 460.90 | 2554.25 | 5791.94 | 19559.48 | ▇▃▁▁▁ |
| distance_to_tertiary_road | 0 | 1.00 | 1259.28 | 1680.04 | 0.02 | 121.25 | 521.77 | 1834.42 | 10966.27 | ▇▂▁▁▁ |
| distance_to_city | 0 | 1.00 | 16663.99 | 10960.82 | 53.05 | 7930.75 | 15030.41 | 24255.75 | 47934.34 | ▇▇▆▃▁ |
| distance_to_town | 0 | 1.00 | 16726.59 | 12452.65 | 30.00 | 6876.92 | 12204.53 | 27739.46 | 44020.64 | ▇▅▃▃▂ |
| rehab_priority | 2654 | 0.44 | 489.33 | 1658.81 | 0.00 | 7.00 | 91.50 | 376.25 | 29697.00 | ▇▁▁▁▁ |
| water_point_population | 4 | 1.00 | 513.58 | 1458.92 | 0.00 | 14.00 | 119.00 | 433.25 | 29697.00 | ▇▁▁▁▁ |
| local_population_1km | 4 | 1.00 | 2727.16 | 4189.46 | 0.00 | 176.00 | 1032.00 | 3717.00 | 36118.00 | ▇▁▁▁▁ |
| crucialness_score | 798 | 0.83 | 0.26 | 0.28 | 0.00 | 0.07 | 0.15 | 0.35 | 1.00 | ▇▃▁▁▁ |
| pressure_score | 798 | 0.83 | 1.46 | 4.16 | 0.00 | 0.12 | 0.41 | 1.24 | 93.69 | ▇▁▁▁▁ |
| usage_capacity | 0 | 1.00 | 560.74 | 338.46 | 300.00 | 300.00 | 300.00 | 1000.00 | 1000.00 | ▇▁▁▁▅ |
| days_since_report | 0 | 1.00 | 2692.69 | 41.92 | 1483.00 | 2688.00 | 2693.00 | 2700.00 | 4645.00 | ▁▇▁▁▁ |
| staleness_score | 0 | 1.00 | 42.80 | 0.58 | 23.13 | 42.70 | 42.79 | 42.86 | 62.66 | ▁▁▇▁▁ |
| location_id | 0 | 1.00 | 235865.49 | 6657.60 | 23741.00 | 230638.75 | 236199.50 | 240061.25 | 267454.00 | ▁▁▁▁▇ |
| cluster_size | 0 | 1.00 | 1.05 | 0.25 | 1.00 | 1.00 | 1.00 | 1.00 | 4.00 | ▇▁▁▁▁ |
| lat_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| lon_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| count | 0 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | ▁▁▇▁▁ |
We can see from the second table of the summary statistics that some of the columns contains many missing values (e.g. Clean_ADM3). Hence, we should remove columns that have many missing values. Also, as seen from the third table, there are also some variables with excessive missing values (e.g. install_year) and these should be removed as well.
We can also notice in the third table, the water_point_population and local_population_11km, have 4 missing values respectively, and they are all referring to the same data records. Hence, we will run the code chunk below to extract the variables used in this exercise, as well as remove the 4 data records that are missing in both the water_point_population and local_population_11km columns. usage_capacity has also been changed to factor class (with only two levels) from numeric class.
Osun_wp_sf_clean <- Osun_wp_sf %>%
filter_at(vars(status, distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean),
all_vars(!is.na(.))) %>%
mutate(usage_capacity = as.factor(usage_capacity))Osun_wp <- Osun_wp_sf_clean %>%
select(c(7,35:39,42:43, 46:47,57)) %>%
st_set_geometry(NULL)Correlation Analysis
We will run the code chunk below to perform correlation analysis.
cluster_vars.cor = cor(Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
lower = "ellipse",
upper = "number",
tl.pos = "lt",
diag = "l",
tl.col = "black")
We can see there is no sign of multi-collinearity in the correlation plot above (i.e. no 2 independent variables have correlation of higher than 0.85).
Building a logistic Regression Model
We will now construct our logistic regression model.
model <- glm(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population+
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = Osun_wp_sf_clean,
family = binomial(link="logit"))We will run blr_regress() to generate the output, based on our code chunk above.
blr_regress(model) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4744 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3887 0.1124 3.4588 5e-04
distance_to_primary_road 1 0.0000 0.0000 -0.7153 0.4744
distance_to_secondary_road 1 0.0000 0.0000 -0.5530 0.5802
distance_to_tertiary_road 1 1e-04 0.0000 4.6708 0.0000
distance_to_city 1 0.0000 0.0000 -4.7574 0.0000
distance_to_town 1 0.0000 0.0000 -4.9170 0.0000
water_point_population 1 -5e-04 0.0000 -11.3686 0.0000
local_population_1km 1 3e-04 0.0000 19.2953 0.0000
usage_capacity1000 1 -0.6230 0.0697 -8.9366 0.0000
is_urbanTRUE 1 -0.2971 0.0819 -3.6294 3e-04
water_source_cleanProtected Shallow Well 1 0.5040 0.0857 5.8783 0.0000
water_source_cleanProtected Spring 1 1.2882 0.4388 2.9359 0.0033
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7347 Somers' D 0.4693
% Discordant 0.2653 Gamma 0.4693
% Tied 0.0000 Tau-a 0.2318
Pairs 5585188 c 0.7347
---------------------------------------------------------------Interpretation of logistic regression
Interpretation should only be done after analysing the respective p-values of each of the variables.
For categorical variables, a positive p-value implies an above average correlation and a negative p-value implies a below average correlation.
For continuous variables, a positive p-value implies a direct correlation and a negative value implies an inverse correlation, while the magnitude of the p-value gives the strength of the correlation.
We can notice in the above summary, there are two variables (distance_to_primary_road & distance_to_secondary_road) with p value greater than the confidence interval of 0.05. Hence, we will exclude these 2 variables as they are not statistically significant.
In the code chunk below, blr_confusion_matrix() of blorr package is used to compute the confusion matrix of the estimated outcomes by using 0.5 as the cutoff value.
blr_confusion_matrix(model, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1301 738
1 813 1904
Accuracy : 0.6739
No Information Rate : 0.4445
Kappa : 0.3373
McNemars's Test P-Value : 0.0602
Sensitivity : 0.7207
Specificity : 0.6154
Pos Pred Value : 0.7008
Neg Pred Value : 0.6381
Prevalence : 0.5555
Detection Rate : 0.4003
Detection Prevalence : 0.5713
Balanced Accuracy : 0.6680
Precision : 0.7008
Recall : 0.7207
'Positive' Class : 1The validity of a cut-off is measured using sensitivity, specificity and accuracy.
Sensitivity: The % of correctly classified events out of all events = TP (TP + FN)
- The value of true positives is higher than the value of true negatives.
Specificity: The % of correctly classified non-events out of all non-events = TN / (TN+ FP)
Accuracy: The % of correctly classified observations over all observations = (TP + TN) / (TP + TN + FP + FN).
False positive rate = FP / (TN + FP)
- We can see that the accuracy here is 0.6739, which is a relatively high value.
Building Geographically Weighted Logistic Regression (gwLR) models
Converting from sf to sp data frame
We will first convert Osun_wp_sf_clean to Spatial Point data frame (Osun_wp_sp). The clean version is used
Osun_wp_sp <- Osun_wp_sf_clean %>%
select(c(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean)) %>%
as_Spatial()
Osun_wp_spclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 11
names : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean
min values : 0, 0.014461356813335, 0.152195902540837, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 1000, 0, Borehole
max values : 1, 26909.8616132094, 19559.4793799085, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 300, 1, Protected Spring Building Fixed Bandwidth GWR Model
Computing fixed bandwidth
bw.fixed <- bw.ggwr(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population+
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = Osun_wp_sp,
family="binomial",
approach="AIC",
kernel="gaussian",
adaptive=FALSE,
longlat=FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2889
1 -2836
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5684.357
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2875
1 -2818
2 -2810
3 -2808
4 -2808
5 -2808
Fixed bandwidth: 59200.13 AICc value: 5646.785
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2847
1 -2781
2 -2768
3 -2765
4 -2765
5 -2765
6 -2765
Fixed bandwidth: 36599.53 AICc value: 5575.148
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2798
1 -2719
2 -2698
3 -2693
4 -2693
5 -2693
6 -2693
Fixed bandwidth: 22631.59 AICc value: 5466.883
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2720
1 -2622
2 -2590
3 -2581
4 -2580
5 -2580
6 -2580
7 -2580
Fixed bandwidth: 13998.93 AICc value: 5324.578
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2601
1 -2476
2 -2431
3 -2419
4 -2417
5 -2417
6 -2417
7 -2417
Fixed bandwidth: 8663.649 AICc value: 5163.61
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2436
1 -2268
2 -2194
3 -2167
4 -2161
5 -2161
6 -2161
7 -2161
8 -2161
9 -2161
Fixed bandwidth: 5366.266 AICc value: 4990.587
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2157
1 -1922
2 -1802
3 -1739
4 -1713
5 -1713
Fixed bandwidth: 3328.371 AICc value: 4798.288
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1751
1 -1421
2 -1238
3 -1133
4 -1084
5 -1084
Fixed bandwidth: 2068.882 AICc value: 4837.017
Iteration Log-Likelihood:(With bandwidth: 4106.777 )
=========================
0 -2297
1 -2095
2 -1997
3 -1951
4 -1938
5 -1936
6 -1936
7 -1936
8 -1936
Fixed bandwidth: 4106.777 AICc value: 4873.161
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2036
1 -1771
2 -1633
3 -1558
4 -1525
5 -1525
Fixed bandwidth: 2847.289 AICc value: 4768.192
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -1941
1 -1655
2 -1503
3 -1417
4 -1378
5 -1378
Fixed bandwidth: 2549.964 AICc value: 4762.212
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1874
1 -1573
2 -1410
3 -1316
4 -1274
5 -1274
Fixed bandwidth: 2366.207 AICc value: 4773.081
Iteration Log-Likelihood:(With bandwidth: 2663.532 )
=========================
0 -1979
1 -1702
2 -1555
3 -1474
4 -1438
5 -1438
Fixed bandwidth: 2663.532 AICc value: 4762.568
Iteration Log-Likelihood:(With bandwidth: 2479.775 )
=========================
0 -1917
1 -1625
2 -1468
3 -1380
4 -1339
5 -1339
Fixed bandwidth: 2479.775 AICc value: 4764.294
Iteration Log-Likelihood:(With bandwidth: 2593.343 )
=========================
0 -1956
1 -1674
2 -1523
3 -1439
4 -1401
5 -1401
Fixed bandwidth: 2593.343 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2620.153 )
=========================
0 -1965
1 -1685
2 -1536
3 -1453
4 -1415
5 -1415
Fixed bandwidth: 2620.153 AICc value: 4761.89
Iteration Log-Likelihood:(With bandwidth: 2576.774 )
=========================
0 -1950
1 -1667
2 -1515
3 -1431
4 -1393
5 -1393
Fixed bandwidth: 2576.774 AICc value: 4761.889
Iteration Log-Likelihood:(With bandwidth: 2603.584 )
=========================
0 -1960
1 -1678
2 -1528
3 -1445
4 -1407
5 -1407
Fixed bandwidth: 2603.584 AICc value: 4761.813
Iteration Log-Likelihood:(With bandwidth: 2609.913 )
=========================
0 -1962
1 -1680
2 -1531
3 -1448
4 -1410
5 -1410
Fixed bandwidth: 2609.913 AICc value: 4761.831
Iteration Log-Likelihood:(With bandwidth: 2599.672 )
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405
Fixed bandwidth: 2599.672 AICc value: 4761.809
Iteration Log-Likelihood:(With bandwidth: 2597.255 )
=========================
0 -1957
1 -1675
2 -1525
3 -1441
4 -1403
5 -1403
Fixed bandwidth: 2597.255 AICc value: 4761.809 bw.fixed[1] 2599.672We can see that the recommended maximum bandwidth is 2,599.672m. We will input this value into the code chunk below to compute the geographically weighted logistic regression.
gwlr.fixed <- ggwr.basic(status ~
distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population+
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = Osun_wp_sp,
bw=bw.fixed,
family="binomial",
kernel = "gaussian",
adaptive=FALSE,
longlat=FALSE) Iteration Log-Likelihood
=========================
0 -1958
1 -1676
2 -1526
3 -1443
4 -1405
5 -1405 gwlr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 21:26:23
Call:
ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road +
distance_to_tertiary_road + distance_to_city + distance_to_town +
water_point_population + local_population_1km + usage_capacity +
is_urban + water_source_clean, data = Osun_wp_sp, bw = bw.fixed,
family = "binomial", kernel = "gaussian", adaptive = FALSE,
longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-124.555 -1.755 1.072 1.742 34.333
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.887e-01 1.124e-01 3.459 0.000543
distance_to_primary_road -4.642e-06 6.490e-06 -0.715 0.474422
distance_to_secondary_road -5.143e-06 9.299e-06 -0.553 0.580230
distance_to_tertiary_road 9.683e-05 2.073e-05 4.671 3.00e-06
distance_to_city -1.686e-05 3.544e-06 -4.757 1.96e-06
distance_to_town -1.480e-05 3.009e-06 -4.917 8.79e-07
water_point_population -5.097e-04 4.484e-05 -11.369 < 2e-16
local_population_1km 3.451e-04 1.788e-05 19.295 < 2e-16
usage_capacity1000 -6.230e-01 6.972e-02 -8.937 < 2e-16
is_urbanTRUE -2.971e-01 8.185e-02 -3.629 0.000284
water_source_cleanProtected Shallow Well 5.040e-01 8.574e-02 5.878 4.14e-09
water_source_cleanProtected Spring 1.288e+00 4.388e-01 2.936 0.003325
Intercept ***
distance_to_primary_road
distance_to_secondary_road
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
water_point_population ***
local_population_1km ***
usage_capacity1000 ***
is_urbanTRUE ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.0 on 4744 degrees of freedom
AIC: 5712
Number of Fisher Scoring iterations: 5
AICc: 5712.099
Pseudo R-square value: 0.1295351
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2599.672
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -8.7228e+02 -4.9955e+00 1.7600e+00
distance_to_primary_road -1.9389e-02 -4.8031e-04 2.9618e-05
distance_to_secondary_road -1.5921e-02 -3.7551e-04 1.2317e-04
distance_to_tertiary_road -1.5618e-02 -4.2368e-04 7.6179e-05
distance_to_city -1.8416e-02 -5.6217e-04 -1.2726e-04
distance_to_town -2.2411e-02 -5.7283e-04 -1.5155e-04
water_point_population -5.2208e-02 -2.2767e-03 -9.8875e-04
local_population_1km -1.2698e-01 4.9952e-04 1.0638e-03
usage_capacity1000 -2.0772e+01 -9.7231e-01 -4.1592e-01
is_urbanTRUE -1.9790e+02 -4.2908e+00 -1.6864e+00
water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01 5.3340e-01
water_source_cleanProtected.Spring -5.2235e+02 -5.5977e+00 2.5441e+00
3rd Qu. Max.
Intercept 1.2763e+01 1073.2154
distance_to_primary_road 4.8443e-04 0.0142
distance_to_secondary_road 6.0692e-04 0.0258
distance_to_tertiary_road 6.6814e-04 0.0128
distance_to_city 2.3718e-04 0.0150
distance_to_town 1.9271e-04 0.0224
water_point_population 5.0102e-04 0.1309
local_population_1km 1.8157e-03 0.0392
usage_capacity1000 3.0322e-01 5.9281
is_urbanTRUE 1.2841e+00 744.3097
water_source_cleanProtected.Shallow.Well 1.7849e+00 67.6343
water_source_cleanProtected.Spring 6.7663e+00 317.4123
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2795.084
AIC : 4414.606
AICc : 4747.423
Pseudo R-square value: 0.5722559
***********************************************************************
Program stops at: 2022-12-17 21:27:36 Here, the AIC value is computed and compared with non-geographic weighted logistic regression AIC value. We can see that the AIC value for geographic weighted logistic regression is lower (4,414.606) in this case. This shows that the model is actually improved since the AIC value has decreased.
Now we will use caret package to analyse the aspects which has resulted in the model’s improvement.
Model Assessment
Converting SDF into sf data frame
To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.
gwr.fixed <- as.data.frame(gwlr.fixed$SDF)Next, we will label yhat values greater than or equal to 0.5 into 1 and else, 0. The result of the logic comparison operation will be saved into a field called most. most is a logical function that has only True or False values, which matches with the observed values.
gwr.fixed <- gwr.fixed %>%
mutate(most = ifelse(
gwr.fixed$yhat >= 0.5, T, F))The code chunk below generates another confusion matrix, which will be used in comparison to the confusion matrix generated at an earlier step (global version).
gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most,
positive="TRUE",
reference = gwr.fixed$y)CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1824 263
TRUE 290 2379
Accuracy : 0.8837
95% CI : (0.8743, 0.8927)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7642
Mcnemar's Test P-Value : 0.2689
Sensitivity : 0.9005
Specificity : 0.8628
Pos Pred Value : 0.8913
Neg Pred Value : 0.8740
Prevalence : 0.5555
Detection Rate : 0.5002
Detection Prevalence : 0.5612
Balanced Accuracy : 0.8816
'Positive' Class : TRUE
As we can see, the structure of both confusion matrix are more or less the same. However, the accuracy has increased to 0.8837, which is a significant increase from the global version of 0.6739. There is no change in the variables, except that one model is non-geographic weighted, and the other is geographic weighted.
The true negatives has also increased from 0.6154 to 0.9005. This means that we can explain the non-functional water points better now. In order to better manage the water points, we should look more into the local neighbourhoods more, rather than at a global scale (i.e. consider local factors instead).
We can thus see that the geographic weighted model is a better model.
Visualising gwLR
Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
select(c(ADM2_EN, ADM2_PCODE, ADM1_EN, ADM1_PCODE, status))gwr_sf.fixed <- cbind(Osun_wp_sf_selected, gwr.fixed)The code chunk below is used to create an interactive point symbol map.
tmap_mode("view")tmap mode set to interactive viewingprob_T <- tm_shape(Osun) +
tm_polygons(alpha=0.1) +
tm_text(text="ADM2_EN") +
tm_shape(gwr_sf.fixed) +
tm_dots(col="yhat",
border.col="gray60",
border.lwd=1) +
tm_view(set.zoom.limits = c(8,14))
prob_TFrom the map above, we can note that some of the states had more clusters of water points than as compared to the others. This is more prevalent in states like Ifelodun, and etc.
# tertiary_TV <- tm_shape(Osun) +
#tm_polygons(alpha=0.1) +
#tm_shape(gwr_sf.fixed) +
#tm_dots(col="distance_to_tertiary_road_TV",border.col="gray60",border.lwd=1) +
#tm_view(set.zoom.limits=c(8,14))# tmap_arrange(tertiary_S,tertiary_TV,asp=1,ncol=2,sync=TRUE)Building Revised Geographically Weighted Logistic Regression (gwLR) models
As we have seen earlier on, there are 2 variables that are not statistically significant. We will proceed to drop these 2 variables and compute the fixed bandwidth geographically weighted logistic regression model.
Converting from sf to sp data frame
We will first convert Osun_wp_sf_clean to Spatial Point data frame (Osun_wp_sp_new). The clean version is used
Osun_wp_sp_new <- Osun_wp_sf_clean %>%
select(c(status,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
is_urban,
water_source_clean)) %>%
as_Spatial()
Osun_wp_sp_newclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 9
names : status, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean
min values : 0, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 1000, 0, Borehole
max values : 1, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 300, 1, Protected Spring Building Fixed Bandwidth GWR Model
Computing fixed bandwidth
bw_new.fixed <- bw.ggwr(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population+
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = Osun_wp_sp_new,
family="binomial",
approach="AIC",
kernel="gaussian",
adaptive=FALSE,
longlat=FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2890
1 -2837
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5681.18
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2878
1 -2820
2 -2812
3 -2810
4 -2810
5 -2810
Fixed bandwidth: 59200.13 AICc value: 5645.901
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2854
1 -2790
2 -2777
3 -2774
4 -2774
5 -2774
6 -2774
Fixed bandwidth: 36599.53 AICc value: 5585.354
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2810
1 -2732
2 -2711
3 -2707
4 -2707
5 -2707
6 -2707
Fixed bandwidth: 22631.59 AICc value: 5481.877
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2732
1 -2635
2 -2604
3 -2597
4 -2596
5 -2596
6 -2596
Fixed bandwidth: 13998.93 AICc value: 5333.718
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2624
1 -2502
2 -2459
3 -2447
4 -2446
5 -2446
6 -2446
7 -2446
Fixed bandwidth: 8663.649 AICc value: 5178.493
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2478
1 -2319
2 -2250
3 -2225
4 -2219
5 -2219
6 -2220
7 -2220
8 -2220
9 -2220
Fixed bandwidth: 5366.266 AICc value: 5022.016
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2222
1 -2002
2 -1894
3 -1838
4 -1818
5 -1814
6 -1814
Fixed bandwidth: 3328.371 AICc value: 4827.587
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1837
1 -1528
2 -1357
3 -1261
4 -1222
5 -1222
Fixed bandwidth: 2068.882 AICc value: 4772.046
Iteration Log-Likelihood:(With bandwidth: 1290.476 )
=========================
0 -1403
1 -1016
2 -807.3
3 -680.2
4 -680.2
Fixed bandwidth: 1290.476 AICc value: 5809.734
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -2019
1 -1753
2 -1614
3 -1538
4 -1506
5 -1506
Fixed bandwidth: 2549.964 AICc value: 4764.056
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2108
1 -1862
2 -1736
3 -1670
4 -1644
5 -1644
Fixed bandwidth: 2847.289 AICc value: 4791.834
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1955
1 -1675
2 -1525
3 -1441
4 -1407
5 -1407
Fixed bandwidth: 2366.207 AICc value: 4755.524
Iteration Log-Likelihood:(With bandwidth: 2252.639 )
=========================
0 -1913
1 -1623
2 -1465
3 -1376
4 -1341
5 -1341
Fixed bandwidth: 2252.639 AICc value: 4759.188
Iteration Log-Likelihood:(With bandwidth: 2436.396 )
=========================
0 -1980
1 -1706
2 -1560
3 -1479
4 -1446
5 -1446
Fixed bandwidth: 2436.396 AICc value: 4756.675
Iteration Log-Likelihood:(With bandwidth: 2322.828 )
=========================
0 -1940
1 -1656
2 -1503
3 -1417
4 -1382
5 -1382
Fixed bandwidth: 2322.828 AICc value: 4756.471
Iteration Log-Likelihood:(With bandwidth: 2393.017 )
=========================
0 -1965
1 -1687
2 -1539
3 -1456
4 -1422
5 -1422
Fixed bandwidth: 2393.017 AICc value: 4755.57
Iteration Log-Likelihood:(With bandwidth: 2349.638 )
=========================
0 -1949
1 -1668
2 -1517
3 -1432
4 -1398
5 -1398
Fixed bandwidth: 2349.638 AICc value: 4755.753
Iteration Log-Likelihood:(With bandwidth: 2376.448 )
=========================
0 -1959
1 -1680
2 -1530
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2376.448 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2382.777 )
=========================
0 -1961
1 -1683
2 -1534
3 -1450
4 -1416
5 -1416
Fixed bandwidth: 2382.777 AICc value: 4755.491
Iteration Log-Likelihood:(With bandwidth: 2372.536 )
=========================
0 -1958
1 -1678
2 -1528
3 -1445
4 -1411
5 -1411
Fixed bandwidth: 2372.536 AICc value: 4755.488
Iteration Log-Likelihood:(With bandwidth: 2378.865 )
=========================
0 -1960
1 -1681
2 -1532
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2378.865 AICc value: 4755.481
Iteration Log-Likelihood:(With bandwidth: 2374.954 )
=========================
0 -1959
1 -1679
2 -1530
3 -1446
4 -1412
5 -1412
Fixed bandwidth: 2374.954 AICc value: 4755.482
Iteration Log-Likelihood:(With bandwidth: 2377.371 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.371 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.942 )
=========================
0 -1960
1 -1680
2 -1531
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2377.942 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.018 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.018 AICc value: 4755.48 bw_new.fixed[1] 2377.371We can see now that the revised fixed bandwidth is 2,377.371m.
gwlr_new.fixed <- ggwr.basic(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
water_point_population+
local_population_1km +
usage_capacity +
is_urban +
water_source_clean,
data = Osun_wp_sp_new,
bw=bw_new.fixed,
family="binomial",
kernel = "gaussian",
adaptive=FALSE,
longlat=FALSE) Iteration Log-Likelihood
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413 gwlr_new.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 21:44:49
Call:
ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city +
distance_to_town + water_point_population + local_population_1km +
usage_capacity + is_urban + water_source_clean, data = Osun_wp_sp_new,
bw = bw_new.fixed, family = "binomial", kernel = "gaussian",
adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-129.368 -1.750 1.074 1.742 34.126
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.540e-01 1.055e-01 3.354 0.000796
distance_to_tertiary_road 1.001e-04 2.040e-05 4.910 9.13e-07
distance_to_city -1.764e-05 3.391e-06 -5.202 1.97e-07
distance_to_town -1.544e-05 2.825e-06 -5.466 4.60e-08
water_point_population -5.098e-04 4.476e-05 -11.390 < 2e-16
local_population_1km 3.452e-04 1.779e-05 19.407 < 2e-16
usage_capacity1000 -6.206e-01 6.966e-02 -8.908 < 2e-16
is_urbanTRUE -2.667e-01 7.474e-02 -3.569 0.000358
water_source_cleanProtected Shallow Well 4.947e-01 8.496e-02 5.823 5.79e-09
water_source_cleanProtected Spring 1.279e+00 4.384e-01 2.917 0.003530
Intercept ***
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
water_point_population ***
local_population_1km ***
usage_capacity1000 ***
is_urbanTRUE ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.9 on 4746 degrees of freedom
AIC: 5708.9
Number of Fisher Scoring iterations: 5
AICc: 5708.923
Pseudo R-square value: 0.129406
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2377.371
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -3.7021e+02 -4.3797e+00 3.5590e+00
distance_to_tertiary_road -3.1622e-02 -4.5462e-04 9.1291e-05
distance_to_city -5.4555e-02 -6.5623e-04 -1.3507e-04
distance_to_town -8.6549e-03 -5.2754e-04 -1.6785e-04
water_point_population -2.9696e-02 -2.2705e-03 -1.2277e-03
local_population_1km -7.7730e-02 4.4281e-04 1.0548e-03
usage_capacity1000 -5.5889e+01 -1.0347e+00 -4.1960e-01
is_urbanTRUE -7.3554e+02 -3.4675e+00 -1.6596e+00
water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01 6.2378e-01
water_source_cleanProtected.Spring -1.3630e+03 -5.3436e+00 2.7714e+00
3rd Qu. Max.
Intercept 1.3755e+01 2171.6373
distance_to_tertiary_road 6.3011e-04 0.0237
distance_to_city 1.5921e-04 0.0162
distance_to_town 2.4490e-04 0.0179
water_point_population 4.5879e-04 0.0765
local_population_1km 1.8479e-03 0.0333
usage_capacity1000 3.9113e-01 9.2449
is_urbanTRUE 1.0554e+00 995.1840
water_source_cleanProtected.Shallow.Well 1.9564e+00 66.8914
water_source_cleanProtected.Spring 7.0805e+00 208.3749
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2815.659
AIC : 4418.776
AICc : 4744.213
Pseudo R-square value: 0.5691072
***********************************************************************
Program stops at: 2022-12-17 21:45:44 We can see that in this scenario, the AIC value has slightly increased to 4418.776 and the AICc value has slightly decreased to 4744.213. The pseudo r-squared value has also decreased slightly to 0.569.
Usually, the performance would have dropped significantly with a reduction in explanatory variables, but it does not happen in this case. Hence, we can prove that the geographically weighted logistic regression model is still a more robust measure, as it did not show much changes in the AIC from the previous geographically weighted logistic regression model, despite having dropped some of the explanatory variables.
Model Assessment
Converting SDF into sf data frame
To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.
gwr_new.fixed <- as.data.frame(gwlr_new.fixed$SDF)Next, we will label yhat values greater than or equal to 0.5 into 1 and else, 0. The result of the logic comparison operation will be saved into a field called most. most is a logical function that has only True or False values, which matches with the observed values.
gwr_new.fixed <- gwr_new.fixed %>%
mutate(most = ifelse(
gwr_new.fixed$yhat >= 0.5, T, F))The code chunk below generates another confusion matrix, which will be used in comparison to the confusion matrix generated at an earlier step (global version).
gwr_new.fixed$y <- as.factor(gwr_new.fixed$y)
gwr_new.fixed$most <- as.factor(gwr_new.fixed$most)
CM_new <- confusionMatrix(data=gwr_new.fixed$most,
positive="TRUE",
reference = gwr_new.fixed$y)
CM_newConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1833 268
TRUE 281 2374
Accuracy : 0.8846
95% CI : (0.8751, 0.8935)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7661
Mcnemar's Test P-Value : 0.6085
Sensitivity : 0.8986
Specificity : 0.8671
Pos Pred Value : 0.8942
Neg Pred Value : 0.8724
Prevalence : 0.5555
Detection Rate : 0.4992
Detection Prevalence : 0.5582
Balanced Accuracy : 0.8828
'Positive' Class : TRUE
As we can see, all 3 models have more or less the structure for the confusion matrix generated. The accuracy in the revised model has increased from 0.8837 to 0.8846, which is a slight increase.
Most of the other values did not vary much from the previous geographically weighted logistic regression model, which shows that overall, the geographically weighted logistic regression model is a very robust model. It’s results do not vary much despite having some of the explanatory variables being dropped.
Visualising revised gwLR
gwr_sf_new.fixed <- cbind(Osun_wp_sf_selected, gwr_new.fixed)The code chunk below is used to create an interactive point symbol map.
tmap_mode("view")tmap mode set to interactive viewingprob_T_new <- tm_shape(Osun) +
tm_polygons(alpha=0.1) +
tm_text(text="ADM2_EN") +
tm_shape(gwr_sf_new.fixed) +
tm_dots(col="yhat",
border.col="gray60",
border.lwd=1) +
tm_view(set.zoom.limits = c(8,14))
prob_T_newAs we can see, some of the water points are still relatively closely clustered, and the results are similar to the ones that are plotted in the previous gwLR model.
A side-by-side comparison would show the similarities.
tmap_arrange(prob_T, prob_T_new, asp=1, ncol=2)Conclusion
As we can see, the geographically weighted logistic regression model is a very robust measure when it comes to modeling the Spatial Variation of the Explanatory Factors of Water Point Status, than as compared to the normal logistic regression model.